7 INTEGRABLE FUNCTIONS AND SETS 123 it follows that n*(\ |/| - \u\ |) ^ e, which by (13.7.2) shows that |/| is integ- rable. Moreover, we have -|/| ^/^ |/| and hence A**(- I/I) = -MI/I) ^ /**(/) = X/) ^ 0*( which establishes (13.7.4.1). Since /+ = K/ + I/I) and/" = i(|/l -/), it follows that/+ and/" are integrable (13.7.3). Iff and # are integrable, then f—gis defined almost everywhere and integrable, and the functions (every- where defined) sup(/, g) and inf(/, g) are equivalent respectively to the func- tions (defined almost everywhere) |(/4- g + |/~ g\) and i(/+ # — |/- #|). Hence they are integrable. (13.7.5) For a function he ^ (resp. g e &*)to be integrable, it is necessary and sufficient that n*(h) < + oo (resp. p*(g) > — oo). For if ju*(/z) < + oo, then for any e > 0 there exists u e JTR(X) such that u £ h and n*(h - M) = ju*(/z) - /*(«) ^ e, by (13.5.1 ) and (13.5.3). The result now follows from the definition of integrable functions. A subset A of X is said to be inferrable if its characteristic function (pA is integrable, or equivalently if ju*(A) and ju*(A) are finite and equal. Their common value J <pA d/j is then denoted by ^u(A) and is called the measure of A. We have /x(0) = 0. The negligible sets are the same as the integrable sets with measure zero. If A is integrable, then so is every set B for which A n (JB and B n (JA are negligible^ and ju(B) = (13.7.6) If A and B are integrable sets, then A u B, A n B and A n (JB are integrable. This follows from the formulas (12.7.3). (13.7.7) Every compact set is integrable. For an open set U to be integrable it is necessary and sufficient that ju*(U) < H-oo. In particular, every relatively compact open set is integrable. This follows immediately from (13.7.5) and (12.7.4). Example (13.7.8) With respect to Lebesgue measure on R, every bounded interval with endpoints a, b is integrable and its measure is \b — a|, by virtue ofion /is integrable if the functions equivalent to /and defined